Adoption Records: Same Total, Different Expression

Decide which expressions always give the same shelter supply totals.

Compare Supply Expressions

Equivalent expressions have the same value for every value of the variable. Select or write each answer.

The shelter packs 7 information sheets for each of n adoptions, then adds 4 extra sheets. Which expression is equivalent to 7n + 4?

A. 11nB. 4 + 7nC. 4n + 7D. 7(n + 4)

Each adoption kit has 3 care cards and 2 contact cards. Select every expression equivalent to 5n, the total number of cards in n kits.

There are 4 supply boxes. Each holds n towels and 6 blankets. Which expression is equivalent to 4(n + 6), the total number of items?

A. 10nB. 4n + 6C. n + 24D. 4n + 24

A record lists n bowls in one storage area and 8n bowls in another. Complete the equivalent expression for the total.

n + 8n = n

A towel inventory is represented by 6n + 18. Sort each expression by whether it is equivalent to this inventory expression.

Check the Inventory Plans

Use the situation to decide which expressions represent the same total for every allowed value of the variable.

For each of n adoptions, Priya packs 2 feeding guides and 5 other care sheets. Omar records the total as 2n + 5n. Which statement correctly compares this with Priya's expression n(2 + 5)?

A. They agree because both totals are always 7n.B. They differ because Priya's total is always n + 7.C. They agree only when there is one adoption.D. They differ because Omar's total is always 5n.

The shelter starts with 5n cans of food. Staff use 2n cans and then receive 9 cans. Which expression gives the same final inventory as 5n − 2n + 9?

A. 3n + 9B. 12nC. 7n + 9D. 3(n + 9)

Each of 3 storage bins contains 2n clean towels and 5 clean blankets. Complete the expression so it is equivalent to 3(2n + 5) for every n.

3(2n + 5) = 6n +

For each of n adoption kits, staff measure 1/2 cup of food in the morning and 1/2 cup in the afternoon. Which expression is equivalent to (1/2)n + (1/2)n, the total cups measured?

A. n + 1B. n/2C. nD. 2n

10.

The shelter has 2 carts. Each cart carries 3n towels and 4 blankets. Select every expression that gives the total number of towels and blankets on both carts.

Test the Evidence

Matching at one value is not enough to prove equivalence. Look for a valid rewrite or a value where the expressions differ.

11.

Two volunteers compare possible expressions for a supply count. Their results are shown below.

Supply Count Checks
n	2(n + 4)	2n + 8	6n
1	10	10	6
2	12	12	12
3	14	14	18

Which conclusion is justified?

A. 2n + 8 and 6n are equivalent because both contain n.B. 2(n + 4) and 6n are equivalent because they match at n = 2.C. 2(n + 4) and 2n + 8 are equivalent by the distributive property.D. All three expressions are equivalent because one row matches.

12.

A volunteer claims that 3(n + 2) and 5n are equivalent expressions for a supply count. Which value of n is a counterexample to the claim?

A. n = 3B. n = 4C. n = 6/2D. n = 9/3

13.

Noah rewrites 5(n + 3) as 5n + 3 on a supply form. Test both expressions at n = 2 to check his rewrite.

When n = 2, 5(n + 3) = , and 5n + 3 = .

14.

Elena records 4(n + 2) care sheets. Malik records 4n + 8 care sheets. Elena says, “Our expressions are equivalent because they both equal 20 when n = 3.” Is her conclusion correct? Is her evidence enough? Explain using the structure of the expressions.